Hadamard Product

Question 1

What is Hadamards product for the completed ζ and L-functions?

Answer 1

There exist constants A and B such that:
ξ(s)=exp(A+Bs)•Π(1-s/ρ)e^(s/ρ)
where the product is over all ρ which are zeros of ξ.

Also, for each Dirichlet character χ mod q, there exist constants Aχ and Bχ such that:

Λ(s,χ)=exp(Aχ+Bχ•s)Π(1-s/ρ)e^(s/ρ)

where now the product is over the zeros of the Λ function.

Question 2

Use the Hadamard product to find an expression for ζ’/ζ(s) and similar for L’/L.
Use the Hadamard product to show that Re(Bχ)=-ΣRe(ρ) where again we sum over roots of Λ

Answer 2

Take logs and differentiate the completed functions in Hadamard form and then compare to normal form.
Evaluate the Λ’/Λ at s=0 to get this constant. Then use the functional equation and look at zeros.

Question 3

Prove the following estimate:
-L’/L(s,χ)=Ο(log(|t|+2)+logq)-
Σ(1/(s-ρ))

Where t is arbitrary with |t|>1 and
-1/2<1

Answer 3

Write:

-L’/L(s,χ)=Ο(1)-L’/L(s,χ)+L’/L(2+it,χ)

and use the big expression we got after playing with the Hadamard product to get:
-L’/L(s,χ)=O(1)-Σ(1/(s-ρ)-1/(2+it-ρ))

Then use a bound on individual parts of this sum and an earlier part of same lemma to show that this has correct order.

Question 4

Give a sharp bound for the number of zeros of ζ and L functions in the box with real part positive less than 1 and imaginary part of magnitude less than T.

Answer 4

As zeros of ζ are symmetric about real axis, define
N(T)=#{ρ:ξ(ρ)=0 and 0=<T}

Then:

N(T)=T/2π•log(T/2π)-T/2π+S(T)/π+O(1)

N(T,χ)=Τ/π•log(qT/2π)-T/2π+O(logqT)

where S(T) is some argument function.